AmatrixSfor all simple current extensions

Transcription

1 IHES/P/96/8 NIKHEF/ hep-th/ January 1996 AmatrixSfor all simple current extensions J. Fuchs, A. N. Schellekens, C. Schweigert Abstract A formula is presented for the modular transformation matrix S for any simple current extension of the chiral algebra of a conformal field theory. This provides in particular an algorithm for resolving arbitrary simple current fixed points, in such a way that the matrix S we obtain is unitary and symmetric and furnishes a modular group representation. The formalism works in principle for any conformal field theory. A crucial ingredient is a set of matrices Sab J,whereJis a simple current and a and b are fixed points of J. We expect that these input matrices realize the modular group for the torus one-point functions of the simple currents. In the case of WZW-models these matrices can be identified with the S-matrices of the orbit Lie algebras that were introduced recently in [1]. As a special case of our conjecture we obtain the modular matrix S for WZW-theories based on group manifolds that are not simply connected, as well as for most coset models. DESY, Notkestraße 85, D Hamburg, Germany NIKHEF, Postbus 41882, NL 1009 DB Amsterdam, The Netherlands. t58@attila.nikhef.nl IHES, 35 route de Chartres, F Bures-sur-Yvette, France

2 1. Introduction One of the more important unsolved problems in conformal field theory is that of classifying and understanding all modular invariant partition functions. Besides the diagonal modular invariant, one can often construct other modular invariant partition functions for conformal field theories. In spite of some recent progress, even in the most extensively studied case of WZW-models based on simple Lie algebras, the classification of these invariants is still incomplete. Moreover, even for the known non-diagonal invariants, a satisfactory interpretation as a full-fledged conformal field theory is available in only a few cases. In this paper, we will be interested in modular invariants that suggest an extension of the chiral algebra, i.e. invariants of the general form N i i l m i,l X l 2. (1.1) Here X l is a character of the original theory (which we will call the unextended theory, even though its chiral algebra will in general itself be an extension of the Virasoro algebra), m i,l a non-negative integer and N i a positive integer. The identity character of the unextended theory appears exactly once (by convention for i = l = 0,with m 0,0 =N 0 =1). Such a partition function suggests an interpretation in terms of an extended algebra, with each term representing the contribution of an irreducible representation of that algebra. The fields which we would like to interpret as the generators of an extended chiral algebra can then be read off the term containing the identity. The existence and uniqueness of such an extended algebra is however by no means guaranteed. Indeed, several examples are known of partition functions of the form (1.1) that do not correspond to any conformal field theory (see e.g. [2,3]). We are not aware of examples of modular invariant combinations of characters of rational conformal field theories that can be interpreted in more than one way in terms of an extended chiral algebra, but this possibility cannot be ruled out either. Having found a modular invariant partition function, the next logical step is to attempt to derive the modular transformation matrix S of the characters of the putative new theory. If such a matrix can indeed be written down, a further important consistency check is the computation of the fusion coefficients using Verlinde s formula [4]. If no inconsistency appears, one can try to compute operator product coefficients and correlation functions. In principle, any of these steps may fail or produce a non-unique answer. In particular we do not study heterotic invariants or fusion rule automorphisms, since our interest is in defining the matrix S for the chiral half of a theory.

3 Apart from a few trivial theories, essentially the only case where the whole programme can be carried through is the extension of WZW-models by currents of spin 1. These invariants can be interpreted as conformal embeddings, and hence the extended theory is again a WZW-model. In this paper we will focus on another case that can be expected to be manageable, namely, for arbitrary rational conformal field theories, the so-called simple current invariants ([5,6], for a review see [7]). These invariants have been completely classified for any conformal field theory [8,9]. Since the construction of the partition function can be formulated in terms of orbifold methods, it is reasonable to expect a conformal field theory to exist. Therefore in particular there should exist a unitary and symmetric matrix S with all the usual properties. Unfortunately, orbifold methods do not seem to be of much help in actually determining this matrix. Such a computation has been carried out so far only for the Z 2 -orbifolds of the c = 1 models [10] and a few other simple examples. Therefore we will follow a different route. Here we will only consider the first step in the programme of describing the (putative) theory which corresponds to a given simple current modular invariant, namely the determination of S. Our current knowledge indicates that for WZW-models based on simple Lie algebras nearly all off-diagonal invariants are simple current invariants. The remaining solutions, which are appropriately referred to as exceptional invariants, are rare (although there are a few infinite series) and unfortunately beyond the scope of this paper. For semisimple algebras far less is known, but certainly the number of simple current invariants increases dramatically [8]. For most of these invariants the modular matrix S, oneof the most basic quantities of a conformal field theory, could not be computed up to now. Although the most important application of our results appears to be in WZW-models, and also in coset theories (see below), we emphasize that simple current constructions are not a priori restricted to WZW-models. For this reason we will set up the formalism in its most general form, and focus on WZW-models only at the end. For simple current invariants there are a few convenient simplifications in (1.1); for example the coefficients m i,l are either 0 or 1, and the vectors m i are all orthogonal. The problem we address in this paper occurs whenever one of the multiplicities N i is larger than 1. This situation occurs if one (or more) of the simple currents in the extension has a fixed point, i.e. if it maps a primary field to itself. If there are no fixed points, one can compute the matrix S simply by looking at the modular transformation properties of the characters. However, if N i > 1forsomevalueofi, this may imply that the new theory has more than one character corresponding to the i th term (the multiplicity will in fact be determined in this paper). In that case all characters in the ith term of the sum (1.1) are identical as functions of the modular parameter τ and possible Cartan angles of the unextended theory, and one cannot disentangle their transformation under τ 1 τ. Fixed points occur very often in simple current invariants. A simple and well-known example is the D-invariant of su(2) level 4, which has the form X 0 + X X 2 2. There are two representations with character X 2. The known modular transformations

4 of su(2) level 4 do not tell us how they transform into each other. Hence we cannot deduce the matrix S directly from that of su(2) level 4. If we assume that a new theory with an extended chiral algebra exists, we know more about S: it must be unitary and symmetric and form, together with the known matrix T, a representation of the modular group, hence satisfy S 4 = 1and(ST) 3 = S 2. In the example the most general form of S that is symmetric and agrees with the known transformations of the su(2) 4 characters is ɛ 1 2 ɛ, ɛ 1 2 +ɛ where ɛ is an unknown parameter. Imposing unitarity fixes ɛ up to a sign. Finally imposing (ST) 3 = S 2 fixes ɛ completely (and one obtains the matrix S of su(3) level 1). It is this solution that we wish to generalize to arbitrary conformal field theories with simple currents. In the general case one can proceed as follows [2]. First one computes the naive matrix S associated with the partition function (1.1) by orbit-averaging the matrix S of the original theory, and by resolving the ith row and column of S into (at most) N i distinct rows and columns. To make the new matrix unitary, correction terms are needed for the entries between fixed point representations. These corrections can be described in terms of a matrix S J that acts only on the fixed points (in fact there is such a matrix for every current J in the extension, hence the upper index). It can then be shown that the resolved matrix S is unitary and symmetric and satisfies ( ST) 3 = S 2 if S J has all those properties on the fixed points. Since T is known and unambiguous, this information can be used in some cases to get plausible ansätze for S J. The problem with this method is that one has to identify the T -eigenvalues of the degenerate representations with a known spectrum. Surprisingly, in many WZW-models these T -eigenvalues can be recognized as those of another WZW-theory (up to an overall phase). In [7] this was achieved for all simple current invariants of WZW-models based on simple, simply laced Lie algebras, as well as for a few other cases. However, the fixed point spectrum obtained for B n and C 2n theories did not correspond (with a few exceptions) to that of a WZW-model or any other known conformal field theory. In addition, the application of this procedure to more complicated combinations of simple currents, with fixed points of all possible types, has never been formulated. The main results reported here are: A conjecture is presented for the matrix S for any simple current invariant of any conformal field theory for which the relevant matrices S J are known. One important problem to be addressed is precisely how many irreducible representations of the extended algebra one gets if N i > 1. We will present a conjecture for this case To prevent confusion between the matrices for the unextended and the extended theories, we denote the former as S and the latter as S.

5 as well; perhaps surprisingly, the answer is not always N i. This means in particular that not even the spectrum of certain extended theories was known before. A matrix S J is presented for any simple current of any WZW-model. This requires the extension of the results of [7] to all simple algebras. The construction of S J was essentially already achieved in [1]. It was found that the missing cases correspond to spectra of twisted affine Kac-Moody algebras. The matrices S J for the missing cases have been obtained earlier from rank-level duality [11], but now for the first time they can be treated on an equal footing for all WZW-models: they can be identified with the modular matrices S of the orbit Lie algebras that are associated to the Dynkin diagram automorphisms induced by the currents J. Although the fixed point resolution matrices S J can in principle be extracted from [1] or [11], we believe it is worthwhile to present the result in a more accessible way. The term conjecture is used in the first item because the conditions we solve are necessary, but not sufficient. An important condition that in the general case is not easy to impose is that the new matrix S must yield sensible fusion rule coefficients when substituted in Verlinde s formula. (Note that a rigorous proof of the conjecture would require in particular an explicit construction of the extended chiral algebra, as well as the proof that it gives rise to a reasonable conformal field theory.) However, there are several reasons why we believe our solution is the correct one, namely: The solution is mathematically natural in the sense that a very simple closed formula can be given that applies to all cases. It has been checked by explicit computation to give non-negative integer fusion coefficients for all types of simple and many semi-simple algebras (of course, such checks have been done only for a limited range of ranks and levels). It can be derived rigorously as the matrix S that describes the transformation of the characters of diagonal coset models. Our results also allow the computation of the matrix S for most coset models. The modular properties of coset models G/H can be described in terms of a formal tensor product of the G-theory with the complement of the H-theory (the complement of a conformal field theory has by definition a complex conjugate representation of the modular group). One gets a matrix S G SH that acts on the branching functions. In many cases some of the branching functions vanish, while others have to be identified with each other, and correspond to a single primary field in the theory. This is known as field identification. Field identification can be formulated as far as modular transformation properties are concerned in terms of a simple current extension of this tensor product, except in a few rare cases (the so-called maverick cosets [12]). Hence the computation of the matrix S of coset models is technically identical to the computation for a suitably chosen integer spin simple current invariant so that our conjecture regarding fixed point resolution for S covers this case as well.

6 However, there is an essential difference in the interpretation and computation of the fixed point characters. In an integer spin modular invariant each of the representations originating from a fixed point has the same character with respect to the chiral algebra of the unextended theory, namely the one appearing within the absolute value symbol in (1.1). On the other hand in coset models the latter character is to be interpreted as the sum of N characters that may be (and in general are) distinct as functions of τ. Hence the degeneracy is lifted, and we can determine S directly from the transformation of the characters. All of this is useful only if one is able to compute the coset characters, which for N>1 are not equal to the branching functions. The differences between the branching functions and the coset characters are called character modifications. We have accomplished this for the diagonal coset models G G/G, by realizing field identification on the entire Hilbert space, and identifying the various eigenspaces of field identification on the fixed points [13]. Having done this, we can prove that for diagonal coset models the character modifications are equal to branching functions of twining characters. Twining characters have been defined in [1] and will be briefly described in section 6. For our present purpose all we need is the fact that in [1] the modular transformations of these characters were obtained. This allowed us to derive the modular transformations of the characters of diagonal coset models. The formula for S we conjecture here is a generalization of the one in [13]. The formula is not identical, since in the general case a complication arises that does not occur for diagonal coset models. While in the case of coset conformal field theories and for integer spin simple current invariants of WZW-theories the associated orbit Lie algebras provide natural candidates for the matrices S J that implement fixed point resolution, it is not clear whether analogous data are available for arbitrary rational conformal field theories. However, we expect that the matrices that describe the transformation of the one-point functions of the simple currents on the torus will do the job. Note that it follows on quite general reasons that these one-point functions have good modular transformation properties [14] and are non-zero only for fixed points. The identification of the matrices S J with the S-matrices for torus one-point functions implies in particular the conjecture that in the case of WZW-models the modular transformation properties of these one-point functions are described by the orbit Lie algebras. Apart from being conceptually elegant, this has the practical advantage that an explicit closed formula for the matrices S J can be given, namely the Kac-Peterson [15] formula of the orbit Lie algebra. The organization of this paper is as follows. In the next section we formulate the conditions we impose on the solution. They consist of six conditions that are beyond question, plus two additional ones that should be considered as working hypotheses. In section 3 we discuss what can be deduced about S using only the six unquestionable conditions. In section 4 we perform a Fourier transformation on the labels of the resolved fixed points. If one imposes the two additional conditions, this suggests an ansatz for the matrix S in the general case, and leads naturally to a definition of the quantities S J.This

7 ansatz is an additional assumption, and for this reason we do not claim to have found the most general solution satisfying all conditions. The characterization of the primary fields of the extended theory and the formula for S, given by equation (5.1), are the main results of this paper. It can be shown to satisfy all conditions given certain properties of the matrices S J. This proof is independent of the heuristic arguments leading us to (5.1), and is briefly summarized in section 5. Readers who are only interested in the result may therefore in fact skip sections 3 and 4. In section 6 we briefly review the concepts of twining characters and orbit Lie algebras and apply our formalism to WZW-models. Realizing that the WZW-model based on G = G/Z, where Gis the universal covering Lie group of G and Z a subgroup of the center of G, is described by the corresponding simple current invariant, this leads in particular to a conjecture for the S-matrix of WZW-models based on non-simply connected compact Lie groups (for the precise definition of these models see [16]). 2. Conditions With respect to the fusion product, the set of simple currents of a conformal field theory forms a finite abelian group, known as the center C of the theory. To any subgroup G C of mutually local integral spin simple currents one can associate a modular invariant partition function in which the chiral algebra is extended by this set of currents. (An explicit expression for the partition function will be given in (2.5).) Our goal is to write down for any such extended theory a pair of matrices S and T, which must satisfy the following requirements: [I] S and T act correctly on the characters. [II] S is symmetric. [III] S is unitary. [IV] S 2 = C. [V] S and T satisfy ( S T ) 3 = S 2. (2.1) Here C is a matrix with entries 0 or 1 satisfying C 2 =1,i.e. a permutation of order 2, which furthermore acts trivially on the identity. The characters of the theory are linear combinations of characters of the unextended theory. This gives us some information about their modular transformations in terms of the matrices S and T of the unextended theory. The meaning of the first condition is that the matrices S and T must reproduce this knowledge. This is the only condition that relates S to the matrix S of the unextended theory. The matrix T follows in a straightforward way from T using condition [I], and therefore we do not specify any explicit conditions for it. As usual, it is a unitary diagonal matrix. Although we do not impose a general integrality condition on the fusion rules derived from S, we make an exception for certain simple current fusion rules, because they are of

8 special importance to us, and can be analyzed. Suppose we are given a simple current J in the unextended theory that is local (i.e., has zero monodromy charge) with respect to all currents in G, so that its orbit is an allowed field in the extended theory, i.e. J is not projected out by the extension. We claim that this orbit gives rise to a simple current in the extended theory. Note that neither the identity primary field nor the simple current J are fixed by the currents in G. It is then easy to see that the S-matrix of the extended theory satisfies S 0,J = S 0,0. It follows that if indeed S leads to correct fusion rules the primary field J in the extended theory has quantum dimension 1 and hence again is a simple current. Therefore we require [VI] Ñ c J,b = δ Jb,c. (2.2) Here Ñ are the fusion coefficients obtained from S via the Verlinde [4] formula, and Jb J b is another primary field in the extended theory, obtained as the fusion product of J and b. The fusion coefficients are finite since S 0,n 0 after fixed point resolution. Conditions [I] [VI] are clearly necessary. It will be helpful to impose two additional working hypotheses, namely [VII] [VIII] Consistency of successive extensions. Fixed Point Homogeneity. (2.3) Condition [VII] applies when there are several distinct paths to the final result. This is the case if there exist several distinct chains of subgroups of the form G H 0 H 1... H n {1}, (2.4) which is possible if the order of G is not prime. If we have a general formula that can deal with any extension, in particular it will give a result for each such chain, when the extensions are performed successively. Condition [VII] states that the answer should not depend on the specific chain chosen. Condition [VIII] means that in the final result the resolved fixed points coming from the same primary field a are indistinguishable in as far as their modular transformations and fusion rules are concerned. This condition has to be handled with some care; while for coset theories it does hold for S and the fusion rules (in all known cases), it does not apply to the characters, for which one has to include so-called character modifications. Note, however, that fixed point resolution might introduce additional simple currents that are not related to simple currents of the unextended theory.

9 An integer spin simple current modular invariant has the general form Z = orbits a Q=0 S a J G/S a X Ja 2. (2.5) Here G is a subgroup of the center whose elements have integer spin; the first sum is over all G-orbits of primary fields in the theory with zero monodromy charge Q. The monodromy charges of primary fields a with respect to the simple current J are defined as the fractional parts Q J (a) =h(a)+h(j) h(ja)mod Z of combinations of conformal weights. simple current orbit are related by The S-matrix elements of fields on the same S La,b =e 2πiQL(b) S a,b. (2.6) The group S a appearing in (2.5), the stabilizer of a, is the subgroup of G that acts trivially on the orbit a; X a is the character of the field a, andx Ja is the character of the representation obtained when the simple current J acts on the orbit a. Since the center C is abelian, all elements of an orbit have the same stabilizer S a. The representations in the orbit a are called fixed points with respect to the currents in the stabilizer. Below we will also use the notation G a := G/S a (2.7) for the factor group of currents that acts non-trivially on a. Implicit in the foregoing discussion is a choice of a representative within each orbit. In the following a,b,c,...always refer to a definite choice of orbit representatives. The primary fields in the unextended theory are then obtained as Ja with J G a. Quantities like S, T and the fusion coefficients N in the extended theory will be distinguished by a tilde. On general grounds one expects [17] that it should be possible to rewrite the invariant (2.5) as a standard diagonal one, i.e. as Z = α X α 2. (2.8) The matrix S acts on the new characters X α. The relation between (2.8) and (2.5) is straightforward if there are no fixed points, i.e. if S a =1foralla. If S a =2or3 there is only one possible interpretation, namely that the orbit a corresponds to precisely S a representations in the extended theory. The characters of those representations are

10 identical with respect to the unextended algebra. If S a 4 there are as many interpretations as there are ways of writing S a as a sum of squares. Each such square can be absorbed in the definition of an extended character X α rather than being interpreted as a multiplicity. In general there is thus ample room for ambiguities. First of all, even the number of primary fields is not evident. Furthermore, for each possible choice of the spectrum (which fixes the matrix T ) there may exist more than one matrix S that satisfies (2.1) and (2.2). 3. Fixed point resolution: Generalities We will now examine the consequences of the first six conditions. The other two will be discussed later. 3.1 Condition [I] Each character X α of the extended theory is in any case a sum of characters of the original theory, which belong to a definite orbit a. There may be more than one character of the extended theory that belongs to the same G-orbit, so we need an extra label. Let us write S a as a sum of squares, S a = i (m a,i ) 2, (3.1) where i labels the different primaries into which a gets resolved (if we also impose condition [VIII], then m a,i has to be independent of i). Corresponding to this definition we have X a,i = m a,i J G a X Ja (3.2) so that i X a,i 2 = S a J G a X Ja 2. Clearly T (a,i),(a,i) = T a,a independent of i. For Swe find X a,i ( 1 τ )=m a,i S Ja,Kb X Kb (τ). K G b J G a b Here and in the rest of this paper we write only the dependence on τ, but there might be additional variables (for example Cartan angles in affine Lie algebra characters). This may in fact be necessary to resolve ambiguities in the unextended theory. The last two sums form together a sum over all fields in the theory, but because of the sum on J only those fields Kb contribute that have zero monodromy charge with respect to all currents

11 in G. We denote this as Q G =0. ForfieldsKb with Q G (Kb) = 0 the matrix element S Ja,Kb is in fact independent of J, soweget X a,i ( 1 τ )=m a,i G a b K G b S a,b X Kb (τ), where we have also used that as a consequence of Q G (a) =0wehaveS a,kb = S a,b. Now we are faced with the problem that in general there is more than one character associated with the orbit whose representative is b. Hence we may write where, in order to satisfy (3.2), K G b X Kb (τ)= 1 N(η, b) η b,j X b,j, j N(η, b) = j η b,j m b,j, and η b,j is a set of coefficients that is present for any b that splits into more than a single representation. We find thus the following formula for S: S (a,i),(b,j) = m a,i G a S a,b 1 N(η, b) η b,j + (a,i),(b,j). Here (a,i),(b,j) is a possible extra term whose presence cannot be inferred from the previous arguments, because of possible degeneracies in the set of characters. (We are assuming here that the set of (generalized) characters of the unextended theory is linearly independent, and we will in any case only consider degeneracies that were introduced by the fixed point resolution.) These degeneracies allow for an additional term, provided it satisfies (a,i),(b,j) m b,j =0. (3.3) j

12 3.2 Condition [II] Now we impose the condition that S must be symmetric. Multiplying this condition with m a,i and summing over i we get 1 G N(η, b) η b,j S a,b + i m a,i (a,i),(b,j) = m b,j G b S a,b. (3.4) Now since S is unitary, for any b there exists an a such that S a,b 0. Let us fix b and pick one such a. Then (3.4) can be solved for η b,j : 1 N(η, b) η b,j = G b G m i b,j m a,i (a,i),(b,j). (3.5) G b S a,b Note that the dependence of the last term on a should cancel. Substituting (3.5) into the formula for S we get G a G S (a,i),(b,j) = m a,i m b b,j S G a,b +Γ (a,i),(b,j), (3.6) where the last term is equal to plus the contribution from the second term in (3.5), Γ (a,i),(b,j) = (a,i),(b,j) m a,i G a G b m a,k (a,k),(b,j). Note that Γ satisfies a sum rule analogous to (3.3). Furthermore symmetry of S implies that Γ must be symmetric. 3.3 Condition [III] The remaining conditions involve a product P of two matrices, either S 2, S S or ( S T ) S. Note that T is constant for fixed a or b. As a consequence, when we write such a product symbolically as P = P S,S + P S,Γ + P Γ,S + P Γ,Γ, then the cross-terms P S,Γ and P Γ,S between the two terms in (3.6) always cancel due to condition (3.3). For P = S S,the term P S,S reads P S,S = b,j m a,im 2 b,j m c,k S a,b S b,c G a G b 2 G c / G 2. The sum over j can be done using (3.1): k P S,S = b m a,i m c,k G a G b G c G S a,b S b,c = b J,K m a,i m c,k G c G S Ja,Kb S Kb,c. Here we have traded the factor G a G b for a sum over the orbits of a and b. Eachtermin these orbits gives the same contribution. The sum on b is over all orbit representatives

13 of Q G = 0 orbits. It can be extended to a sum over all orbits because the contributions of the Q G 0 orbits cancel among each other owing to the sum on J. Together with the sum on K we now have a sum over all primary fields in the unextended theory, and we can use unitarity (respectively S 2 = C, orsts = T 1 ST 1 ) in the unextended theory. Requiring unitarity of S leads then to the condition Γ (a,i),(b,j) Γ (b,j),(c,k) = δ ac b,j ( δ ik m ) a,i m a,k. S a Note that the right hand side is a projection operator, Pik a δ ik m a,i m a,k. S a A special case of this result was already obtained in [2], but there all multiplicities m a,i were assumed to be equal to Condition [IV] The computation for S 2 yields in a similar way the relation b,j Γ (a,i),(b,j) Γ (b,j),(c,k) = C (a,i),(c,k) m a,i m c,k S c C Ja,c, J where C (a,i),(c,k) is the charge conjugation matrix of the extended theory and C a,c that of the unextended one. The sum in the second term can only contribute if a and c are representatives of conjugate orbits, and in that case it contributes 1, and otherwise 0. We may thus introduce a matrix Ĉa,c on orbit representatives which is 1 if a is conjugate to some field on the G-orbit of c, and 0 otherwise. Then we get b,j Γ (a,i),(b,j) Γ (b,j),(c,k) = C (a,i),(c,k) m a,i m c,k S c Ĉ ac. Using the sum rule analogous to (3.3) that is valid for the matrix Γ, we conclude that k C (a,i),(c,k) m c,k = m a,i Ĉ a,c. This implies that C (a,i),(c,k) can only be non-zero between orbits a and c with Ĉa,c = 1. Furthermore, conjugate fields must have the same value of m. It follows that the set of numbers m i must be identical on conjugate orbits. Hence

14 we may write C (a,i),(c,k) = Ĉ a,c C c i,k, (3.7) where Ci,k c is a conjugation matrix that is introduced by the fixed point resolution. Because C and Ĉ are symmetric, the matrices Cc must satisfy C c =(C c ) T if c is the conjugate of c. The final result is therefore Γ (a,i),(b,j) Γ (b,j),(c,k) = Ĉa,c Ci,l c P l,k c. b,j l 3.5 Condition [V] Condition [V] is most conveniently dealt with in the equivalent form S T S = T 1 S T 1. For S T S we find Γ (a,i),(b,j) Tb,b Γ (b,j),(c,k) =( T 1 S T 1 ) (a,i),(c,k) b,j J m a,i m c,k G c G T 1 Ja,Ja S Ja,c T 1 c,c. The sum on J just yields a factor G a, and then the last term cancels the first contribution from S. The result is Γ (a,i),(b,j) T b,b Γ (b,j),(c,k) =( T 1 Γ T 1 ) (a,i),(c,k). b,j 3.6 Some remarks on fusion rules Although it seems to be quite difficult to examine the fusion rules in general, we can discuss the case that one of the three fields is not a fixed point (if even fewer fields are fixed points, the discussion is completely straightforward). Note that the formula i m a,iγ (a,i),(b,j) = 0 implies that Γ = 0 if a field is resolved into only one primary field of the extended theory. In particular, there are no correction matrices Γ for fields that are not fixed points.

15 We obtain the following formula for the fusion coefficients: (c,k) Ñ = G b G c a,(b,j) G 2 m b,j m c,k N J G c a,jb + d,l S a,d S 0,d Γ (b,j),(d,l) Γ (c,k),(d,l). In principle the requirement that the coefficient should be a positive integer imposes restrictions on Γ, but these conditions do not look particularly useful. This is even more true for the fusion of three resolved fixed points. A few things can be learned, though. Multiplying with m c,k and summing over k we get Ñ k (c,k) a,(b,j) m c,k = m b,j Na,Jb c. J G b This has a few implications. If a b contains terms in the orbit of c, thenñ a,(b,j) cannot be zero for all k. Furthermore, if a b does not yield any contribution in the c (c,k) orbit, then Ñ = 0. Thus the new fusion rules must respect the orbit-orbit maps a,(b,j) of the original fusion rules, although the distribution of fields may be non-trivial. We can in fact say more. If n = J G b Na,Jb =1and G b = G c,thenitcan be shown that the vector m c is a permutation of m b. Since this is not a surprising result, we omit the details of the proof. The condition that n =1and G b = G c is in particular satisfied if a is a simple current, but in general this is by no means the only possibility. If there is any non-fixed point field a that maps orbit b to c with multiplicity 1, then orbit b and c must have the same decomposition vector m (if G b = G c ). The result indicates that fixed points which have the same stabilizer should also possess the same decomposition m; it is difficult to imagine how a different decomposition could still provide a solution to all constraints. 3.7 Condition [VI] Suppose there is a simple current L in the theory which is local with respect to all currents by which we extend the algebra. Then according to the remarks before equation (2.2) L will again be a simple current in the extended theory. (Note that, as before, L stands for a definite representative of the G-orbit of the additional currents.) Hence L must act as a simple current on the resolved fixed points. Thus if in the unextended theory L a = b, then as seen above we have in the extended theory c (c,k) L (a, i) =(b, j)

16 for some j. Hence we have both Q L (a) =h(a)+h(l) h(b)mod1 (3.8) and Q L ((a, i)) = h((a, i)) + h(l) h((b, j)) mod 1. Since the respective conformal weights are the same up to integers, we see that on all fields Q L = Q L.NotethatLwas assumed to be local with respect to G, sothath(l)is a constant (modulo integers) on the G-orbit of L, and hence the notation makes sense. To relate the matrix element Γ (a,i),(c,k) to Γ (b,j),(c,k), we recall the relation (2.6) that a simple current L imposes on the S-matrix elements. Combining this formula with the analogous relation S L(a,i),(b,j) =e 2πiQL(b) S(a,i),(b,j) (3.9) for S, we obtain an analogous relation for Γ: Γ L(a,i),(c,k) =e 2πiQL(c) Γ (a,i),(c,k). (3.10) 4. Fourier decomposition Suppose we consider an arbitrary fixed point resolution, where a fixed point a is split into M a primary fields. From now on we will impose the homogeneity condition [VIII], and therefore in particular we will only consider the case that a is split into M a primary fields with identical multiplicity factors m a,i = m a.then S a =(m a ) 2 M a. (4.1) Suppose by some as yet unspecified procedure we obtain a matrix S (a,i),(b,j) satisfying all the requirements listed in section Group characters For each fixed point choose an abelian discrete group M a with as many characters as there are resolved fields, i.e. M a = M a. Later we will identify this group as a subgroup of the stabilizer, but for the moment there is no need to be specific. An important role

17 will be played by the group characters Ψ a i, i =1,2,...,M a,ofm a. The characters are a complete set of complex functions on the group satisfying Ψ a i (g)ψ a i (h) =Ψ a i(gh), Ψ a i (g 1 )=Ψ a i(g), (4.2) Ψ a i (1)=1, for all g, h M a (1denotes the unit element of M a ). For these characters the orthogonality and completeness relations Ψ a i (g)ψ a i (h) = M a δ gh, i Ψ a i (g)ψ a j (g) = M a δ ij (4.3) g hold. For cyclic groups Z N we will label the elements by integers 0 g < N; the characters read Ψ a l (g) =e2πilg/n for 0 l<n. The groups M a are chosen isomorphic on conjugate G-orbits, as well as on G-orbits connected by any additional simple currents. This is possible since we have seen that the decomposition vector m is preserved by charge conjugation and simple current maps. We define the Fourier components of S with respect to the groups M a as S g,h a,b := 1 Ψ a i Ma M (g) Ψ b j (h) S (a,i),(b,j). b i j The inverse of this transformation is S (a,i),(b,j) = 1 Ψ a i (g)ψ b Ma M j(h) S g,h a,b. (4.4) b g h Now we will examine the implication of conditions [I] [V] in terms of the Fourier components.

18 4.2 Condition [I] Because of this condition some of the elements S g,h a,b are already known. According to the general expression (3.6) for fixed point resolution, we have S (a,i),(b,j) = m a m b G S a S b S a,b +Γ (a,i),(b,j). (4.5) Using (4.5), we can compute S g,h for g = 1 (or h = 1). The characters obey Ψ a i (1) =1 for all i. Using also m a,i Γ (a,i),(b,j) =0 i and the fact that the multiplicities are by assumption independent of i, it follows that in this case Γ does not contribute. The only contribution is thus S 1,h a,b = 1 Ma M b i j Ψ b j(h) G m am b S a S b S a,b. Because of the orthogonality relation of the characters, this vanishes unless h = 1. For h=1the sums over i and j yield M a M b, and the result is S 1,h a,b = Sh,1 a,b = G m am b Ma M b δ h,1 S a,b = S a S b G Sa S b δ h,1 S a,b. (4.6) 4.3 Condition [II] Symmetry of S (a,i),(b,j) implies S h 1,g 1 b,a = S g,h a,b. 4.4 Condition [III] Unitarity of S can be shown to be equivalent to h,b S g,h a,b (Sh 1,f b,c ) = δ ac δ g,f 1. (4.7)

19 4.5 Condition [IV] Consider now the product S 2. Using (3.7) we find S g,h a,b Sh,f b,c = 1 Ψ a i Ma M (g) Ψ c l (f) Ĉ a,c Ci,l c. c h,b i l This vanishes unless c = a.ifc=a,thenm a =M c, and we may write the result as h,b S g,h a,b Sh,f b,c = Ĉa,c 1 Ψ c M i(g) Ψ c l (f) Cc i,l. c i,l If Ci,l c = δ il the result is Ĉ a,c δ gf. Otherwise Ci,l c the characters. We may define defines a permutation of the labels i of C c g,f := 1 Ψ c M i(g) Ci,l c Ψc l (f), c i,l so that the result is h,b S g,h a,b Sh,f b,c = Ĉa,c C c g,f. The matrix Cg,f c is unitary, but in general it is not a permutation even though the Fourier transform Ci,l c is. 4.6 Condition [V] A completely analogous computation can be done for the relation ( S T ) 3 = C of the modular group. The result is h,b S g,h a,b T b,b Sh,f b,c = T 1 a,a Sg,f a,c T 1 c,c.

20 4.7 Condition [VI] If the extended theory has a surviving simple current L we have S L(a,i),(b,j) =e 2πiQL(b) S(a,i),(b,j). (4.8) The action of L on the resolved fixed point moves the orbit representative a to La, and the label i to Li. Here Li denotes some other label of the resolved fixed points of the field La. Expanding the left and right hand side of (4.8) into Fourier modes, we get e 2πiQL(b) S g,h a,b = M 1 a i,f Ψa i (g) ΨLi a (f) Sf,h La,b. Analogously as we did above for charge conjugation, we define a matrix Fg,f a (L) := 1 Ψ a M i(g) ΨLi(f) a, a i so that we can write the result as e 2πiQL(b) S g,h a,b = f Fg,f a (L) Sf,h La,b. (4.9) 4.8 Condition [VII]: Successive extensions Condition [VII] has several consequences. We will first compare an extension in two steps with a complete extension, i.e. in the notation of (2.4) we compare G H {1} with G {1}. For simplicity we consider only the case where p G / H is prime, which by recursion includes all other cases anyway. Performing the extension by H, we obtain a modular matrix S (a,i),(b,j),whichcan be described by Fourier components S g,h a,b. By assumption the extended theory has an integral spin simple current L of prime order p. When we further extend by this current L, the stabilizer of any field a remains either unchanged or is enlarged from Sa H to Sa G SH a. If it remains unchanged, then for L G\Hwe have L(a, i) =(b, j) with a b, and hence L has in any case no fixed points; this situation requires no further discussion. On the other hand, if the stabilizer is enlarged, then Sa G = p Sa H. Now two cases have to be distinguished: Case A: L(a, i) =(a, i). Case B: L(a, i) =(a, Li) withli i.

21 In case A a fixed point resolution is necessary for the primary field labelled by (a, i), whereas in case B a field identification takes place. Now condition [VIII] implies immediately that any fixed point of a simple current of prime order must be resolved into p fields with multiplicity m α = 1; hence in case A the field (a, i) isresolvedintopnew primary fields (a, i, α), α =1,2,...,p. The field identification in case B combines p primaries (a, i) (withfixeda, but distinct values of i) into a single primary field of the G-extended theory. In other words, the Ma H fields (a, i) with given a are combined into Ma H /p new fields (a, i l ), where i l (l =1,2,...,Ma H /p) denotes some definite choice among the labels i, reducing the label set by a factor p. It follows that if all extensions were as in case A, the multiplicities m would always be equal to 1, and the total number of fields would be equal to S a, the order of the stabilizer of a. In contrast, case B amounts to a reduction of the number of primary fields by a factor p 2, which is accompanied by an enlargement of the multiplicity m by a factor of p due to the sum over the L-orbit. (Note that L must generate an orbit of order p on the resolved H-fixed points, and under condition [VIII] this is only possible if H contains a factor p.) As a consequence, the number of primaries into which a fixed point a of the extension by G is resolved can in general be any integer S a /N 2 with N Z. Inspecting successive extensions allows us to decide which of these possibilities is realized. Let us now first consider case A. For the Fourier basis we may take, without any loss of generality, a subgroup M H a M G a with M H a = M G a /p. Then any g M G a can be written in the factorized form g = hw l,whereh M H a,andwherewis a coset representative of a non-trivial element of the coset M G a /M H a (wetaketheidentity 1 M G a as the representative for the trivial element of MG a /MH a ); thus wl M H a for 0 <l<p. The M G a -characters then act as Ψ a (i,α) (hwl )=Ψ a (i,α) (h)ψa (i,α) (wl )=Ψ a i(h)ψ a (i,α) (wl ). Here in the last equality we have used the fact that for h M H a the characters satisfy Ψ (i,α) (h) =Ψ i (h), where the latter are characters of M H a. We can now write down formulas for the first extension, the second extension, and the full extension. The non-fixed orbits can be taken into account by restricting the label α to a single value. To make the notation unambiguous, resolved matrices will now be distinguished by a superscript G and H (instead of a tilde), and other quantities are labelled in the same way. For the first extension we have S(a,i),(b,j) H = 1 Ma HM b H and for the second extension g M H a h M H b g,h ;H Sa,b Ψ a i (g)ψ b j(h), (4.10) S G (a,i,α),(b,j,β) = M H a M H b M G a M G b 0 n<p 0 m<p S n,m (a,i),(b,j) Ψa α (n)ψb β (m). (4.11)

22 Note that the second extension requires a Z p -Fourier transform even if M G is not the direct product of M H by Z p. For the full extension we find S G (a,i,α),(b,j,β) = 1 Ma G M G b g M G a h M G b Furthermore we have the relations (cf. (4.6)) 1,1 ;H Sa,b = g,h ;G Sa,b Ψ a (i,α) (g)ψb (j,β) (h). (4.12) H S H a S H b S a,b (4.13) and 1,1 ;G Sa,b = G S a,b, S 1,1 Sa G Sb G (a,i),(b,j) = G H S H a S H b S G a S G b SH (a,i),(b,j). (4.14) Using these identities it is not difficult to show that g,h ;G Sa,b = G Sa H SH b ;H H Sa G Sb G Sg,h a,b (4.15) when g M H a and h MH b. Furthermore, if g MH a but h MH b (or vice versa), g,h ;H then S must vanish. a,b In case B we have M H a MG a, and we can work with group characters Ψa (i) acting on a M H a, where the labelling is such that the subset Ψa (i p) forms a set of characters of M G a. Again we will include the limiting case M G = M H, to allow also for fields that are not fixed points. Of course, the formula for the first extension remains the same as in case A. For the full extension we have S G (a,i = 1 p),(b,j p) Ma G M G b and for the second extension g M G a h M G b g,h ;G Sa,b Ψ a i p (g)ψ b j p (h), (4.16) S G (a,i = G Sa H Sb H p),(b,j p) H Sa G S G b SH (a,i. (4.17) p),(b,j p) This is just (3.6) with m i = 1 and Γ = 0, since in the second step there are no fixed points to resolve. Note that all matrix elements of S(a,i H p),(b,j p) are the same on the L-orbits, so that the answer does not depend on which element we select for i p.

23 Combining this information, we obtain again the result (4.15), except that this time it completely determines S g,h ;G, i.e. is valid for all g M H a and all h MH b. In principle it might happen that cases A and B occur simultaneously for a given extension. Then there are matrix elements of S G between fields a and b with M G a M H a and M G b MH b. The analysis of this mixed case is essentially the same as in the previous case, and the result is once again (4.15), with the restriction that S g,h = 0 whenever h M H b. All these cases are summarized by the formula S G (a,i),(b,i ) = 1 Ma G M G b g M G a h M G b g,h ;G Sa,b Ψ a I (g)ψb I (h) (4.18) for the full extension, where I now stands for either the combination of labels (i, α) or the single label i p, or just the label i or α, and analogously for I. Furthermore we always have the relation (4.15). In case A this only gives us part of S g,h ;G whereas for case B it gives us all of S g,h ;G. Note that even though (4.18) is universal, the factor 1/ Ma G for case B is by a factor p larger than in case A. 4.9 Condition [VII]: Commutativity of extensions In the previous section we have analyzed the consequences of condition [VII] by comparing two successive extensions to the full extension. Another aspect of condition [VII] is that two successive extensions should commute whenever each of them can be performed as the first extension. To check this commutativity, we have to compare the embedding chains G = H 1 H 2 H 1 {1} and G = H 1 H 2 H 2 {1}. Consider two fields a and b with stabilizer S = S 1 S 2, all with implicit labels a respectively b. For simplicity we will assume that case A applies in all cases, so that M = S. Without loss of generality we may then choose for the group M i the stabilizer S i. Requiring that each of the two embedding chains yields the same answer leads to the condition S g,h ;G = G Si H SH i H i S G S G Sg,h ;H i (4.19) when either g or h are restricted to the subgroup M i. In particular S g,h ;G vanishes when g and h are from different factors of G. This holds equally well for any other decomposition of G.

24 4.10 The matrices S J At this point it is worth stressing that so far there was no need to specify the discrete abelian group M a (except for the restriction that the groups associated to successive embeddings are contained in each other). Rather, choosing a particular group M a is merely a matter of convenience. However, while for any fixed point resolution the Fourier transformation (4.4) can be performed for any arbitrary choice of M a, this manipulation is not likely to lead to useful results unless a clever choice of M a is made. Now the results of the previous sections inspire us to make the following ansatz for S g,h a,b. First of all, we identify the elements g and h of the groups M a and M b with elements J and K of the stabilizer S a or S b, respectively. This is obviously possible if all multiplicities m a,i are equal to 1, since in that case M a = S a. Otherwise these numbers differ by a square (because of condition [VIII]), and one can always find a subgroup of S a that has the right size. This subgroup may not be unique, but we will soon make a canonical choice. Now we make the ansatz J,K ;G Sa,b := G Sa S b δ JK S J a,b. (4.20) This defines a matrix Sa,b J for each current, which is however independent of the extension one considers. The precise definition of S J can already be obtained by considering the minimal extension for which it appears, namely the extension by J itself (or more precisely, the discrete group H J it generates). In that case (4.20) reads J,K ;HJ Sa,b = δ J,K Sa,b J, since G = H J, and the identification of J with an element of S a and one of S b means that S J ab = 0 unless Ja = a and Jb = b.thusj S a S b, which implies that H J = S a = S b. Note that for G =(Z 2 ) n this structure follows completely from the foregoing discussion. For each single H J = Z 2 extension by a current J we have S 0,J =0=S J,0 (4.21) In principle it could happen that J S, but J M. Since the Fourier transforms are defined using M, this would imply that Sa,b J is then not defined for all primary fields a. Itcanbeshown (see the appendix) that this situation which can only be due to successive resolutions according to case B above can never arise within a single cyclic group. (Also, one can argue that these matrix elements are anyway never needed in further extensions.) Hence if we extend the algebra only by J (and its powers), then all fixed points are fully resolved into fields that can be described in terms of S a.

25 and S 0,0;HJ a,b = H J S HJ a S HJ b S a,b, J,J ;HJ Sa,b = H J S HJ a S HJ b S J a,b (= SJ a,b ). (4.22) The last of these equalities is the definition of S J, while the others follow from (4.6) and tell us that Sa,b 1 = S a,b. Using (4.19), this implies immediately that (4.20) holds for any further extension. Therefore the non-trivial assumption in (4.20) is that the matrices S J,K vanish for distinct currents J and K even if they belong to the same cyclic subgroup of G and have the same order. Now we substitute the ansatz (4.20) in conditions [II] [V], derived for the most general form of the resolution procedure. We consider first the defining matrices obtained for the minimal extension. We find: Condition [II]: Condition [III]: b S J a,b S J a,b = SJ 1 b,a. (4.23) (SJ 1 b,c ) = δ bc. When combined with (4.23), this implies that S J is unitary. Condition [IV]: We obtain b S a,b J SJ b,c δ JK = Ĉ a,c CJ,K c,wherecc J,K is a unitary matrix. The condition [IV] says that it must also be diagonal, so that its diagonal matrix elements must be phases. Hence we get Sa,b J SJ b,c = Ĉa,c ηc J b with some phases ηc J. The fact that CJ,K c is diagonal implies that 1 Ma i,j Ψa i (J)Cc i,j (J)Ψa j (K) = ηc J δ JK. Multiplying with Ψ a k (K) and summing over K then gives Ψ a i (J) Cc i,k = ηj c Ψa k (J). i In the sum on the left hand side only a single term survives. Define k c by C c j,k = δ j,k c.